MS-DS Master of Data science Statistical Inference Concepts 2 — Questions and Answers
Question 1: A 95% confidence interval for the mean is [42, 58]. What is the correct interpretation?
- There is a 95% probability the true mean is between 42 and 58
- If we repeated the sampling procedure many times, 95% of such intervals would contain the true mean (Correct answer)
- 95% of the data falls between 42 and 58
- The sample mean has a 95% chance of equaling the true mean
Correct answer: If we repeated the sampling procedure many times, 95% of such intervals would contain the true mean
A confidence interval is a procedure: 95% of intervals constructed this way will capture the true parameter, not a probability statement about one specific interval.
Question 2: Which of the following is a consequence of increasing the sample size in hypothesis testing while keeping α fixed?
- Type I error rate increases
- Type II error rate decreases (power increases) (Correct answer)
- The critical value for the test statistic increases
- The p-value threshold changes
Correct answer: Type II error rate decreases (power increases)
Larger samples reduce variance, making the test more sensitive and increasing statistical power (decreasing Type II error).
Question 3: In a two-sided t-test with H₀: μ = 10 and a test statistic t = 2.1 (df=25), the p-value is approximately 0.046. At α = 0.05, what is the conclusion?
- Fail to reject H₀ because 2.1 > 0.05
- Reject H₀ because p < α (Correct answer)
- Reject H₀ because the test statistic is positive
- Fail to reject H₀ because df = 25 is too small
Correct answer: Reject H₀ because p < α
Since p ≈ 0.046 < 0.05 = α, we reject the null hypothesis at the 5% significance level.
Question 4: The Neyman-Pearson lemma establishes that the likelihood ratio test is:
- Uniformly most powerful for all alternative hypotheses
- The most powerful test for a simple null versus a simple alternative (Correct answer)
- Always equivalent to the chi-square test
- Valid only for large samples
Correct answer: The most powerful test for a simple null versus a simple alternative
The Neyman-Pearson lemma proves that the likelihood ratio test is the most powerful test for comparing two simple (point) hypotheses.
Question 5: A researcher uses bootstrapping to estimate the standard error of the median. What does this method fundamentally rely on?
- The median being normally distributed
- Resampling with replacement from the observed data to approximate the sampling distribution (Correct answer)
- A known parametric form for the population distribution
- The central limit theorem applying to medians
Correct answer: Resampling with replacement from the observed data to approximate the sampling distribution
Bootstrapping treats the empirical distribution as a proxy for the population and resamples with replacement to build an approximate sampling distribution.
Question 6: Which property ensures that a maximum likelihood estimator (MLE) achieves the Cramér-Rao lower bound asymptotically?
- Sufficiency
- Consistency
- Asymptotic efficiency (Correct answer)
- Unbiasedness
Correct answer: Asymptotic efficiency
MLEs are asymptotically efficient, meaning their variance achieves the Cramér-Rao lower bound in the limit of large samples.
Question 7: In Bayesian inference, the posterior distribution is proportional to:
- The likelihood only
- The prior only
- The likelihood times the prior (Correct answer)
- The marginal likelihood divided by the prior
Correct answer: The likelihood times the prior
By Bayes' theorem, P(θ|data) ∝ P(data|θ) × P(θ), i.e., posterior ∝ likelihood × prior.
